OGLE-2016-BLG-0168 Binary Microlensing Event: Prediction and Confirmation of the Microlens Parallax Effect from Space-based Observations

The event was announced by the Optical Gravitational Lensing Experiment (OGLE: Udalski et al. 2015a) based on observations with its 1.3 m Warsaw telescope with a $1.4\,{\deg }^{2}$ camera located at the Las Campanas Observatory in Chile. The event was alerted by the Early Warning System (Udalski et al. 1994; Udalski 2003) of the OGLE survey on 2016 February 21. The OGLE data in the I-band were reduced by a pipeline based on the difference-imaging analysis method (Alard & Lupton 1998; Wozniak 2000). The uncertainties of the OGLE data were re-scaled according to the description in Skowron et al. (2016).

The Korea Microlensing Telescope Network (KMTNet: Kim et al. 2016) survey, which is designed for high-cadence monitoring toward the galactic bulge with a large field-of-view, independently observed the event. KMTNet is a telescope network consisting of three identical 1.6 m telescopes with $4\,{\deg }^{2}$ cameras located at the Cerro Tololo Inter-American Observatory in Chile (KMTC), the South African Astronomical Observatory in South Africa (KMTS), and the Siding Spring Observatory in Australia (KMTA). For the event, KMTC and KMTA observations covered the caustic entrance ($\mathrm{HJD}-2450000=\mathrm{HJD}^{\prime} \sim 7474.2$) and exit ($\mathrm{HJD}^{\prime} \,\sim 7532.5$) parts of the light curve with a ∼15 minute cadence. KMTNet data in the I-band were reduced by pySIS (Albrow et al. 2009), which employs the image subtraction method.

The event was also observed in both the I and H bands by the SMARTS 1.3 m telescope at CTIO in Chile. These data were not used in the modeling, but were used to determine the $(I-H)$ source color (see Section 3.3).

The event was observed by the Spitzer space telescope with the 3.6 μm channel (hereafter, L-band) of the IRAC camera. Briefly, the event was selected on 2016 June 16 as a subjective target based on the selection criteria described in Yee et al. (2015a) because the light curve from ground-based observations showed typical anomaly features caused by the binary lens system. The observations started on 2016 June 18 ($\mathrm{HJD}^{\prime} \sim 7557.93$) and ended July 14 ($\mathrm{HJD}^{\prime} \sim 7584.48$). During 4 weeks of observations with cadence $\sim 1\,{\mathrm{day}}^{-1}$, 28 data points for the event were gathered and then the data were reduced using methods described in Calchi Novati et al. (2015b). In the Spitzer observations, there exist sinusoidal trends whose origins are unknown. However, we can securely determine the modeling parameters in this case despite the trend in the Spitzer data. Indeed, several publications used Spitzer data (which also had similar photometric systematics) to conclude that the systematics in Spitzer data are not likely to be an issue for determining modeling parameters (e.g., Poleski et al. 2016; Shvartzvald et al. 2017; Zhu et al. 2017).

The source star of the event is located in a severely extincted field. The A_I value is measured from the color–magnitude diagram (CMD) analysis of this event (see Section 3.3). Based on the I-band extinction, the A_L value is calculated using the relationship between optical and infrared extinction (Cardelli et al. 1989). The source extinction is ${A}_{I}\sim 4.9$ in the I-band and ${A}_{L}\sim 0.35$ in the L-band. As a result, the source is relatively faint for ground-based observations from OGLE and KMTNet. In contrast to ground-based observations, the source is a quite bright target for Spitzer observations.

In Figure 1, we present observed light curves as seen from ground and space. The light curve shows a typical "U"-shape of a binary lensing light curve. As shown in the zoom-ins, the caustic entrance and exit are well-covered by the KMTNet survey, thus we can clearly measure the angular Einstein ring radius. In addition, the time between the caustic entrance and exit is ∼60 days. This is long enough to expect to detect signals in the ground-based light curve caused by the annual microlens parallax and lens-orbital effects. In Figure 2, we also present the geometry of the event, which is described in detail in Sections 4.1 and 4.3.

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In Table 1, we present models with the best-fit parameters of the various solutions considering the lens-parallax and lens-orbital effects. We find that there exist two degenerate models, $({u}_{0}\lt 0)$ and $({u}_{0}\gt 0)$, for the ground-based light curve. For both cases, we consider the microlens parallax and the lens-orbital effects. When the microlens parallax effect (annual microlens parallax effect: Gould 1992) is introduced, we find that additional ${\chi }^{2}$ improvements compared to the static model are 159.4 and 152.6 for the $({u}_{0}\lt 0)$ and $({u}_{0}\gt 0)$ cases, respectively. In addition, when we supplement the lens-parallax model with the lens-orbital effect (approximated lens-orbital effect: Shin et al. 2013), we find that the ${\chi }^{2}$ improvements are 115.0 and 108.7 for the $({u}_{0}\lt 0)$ and $({u}_{0}\gt 0)$ cases, respectively. This implies that both effects are clearly detected for both cases. Since we found significant ${\chi }^{2}$ improvement when we considered the lens-orbital effect, we investigated complete Keplerian orbital solutions (with parameters adopted from Shin et al. 2011; Skowron et al. 2011). However, the Keplerian orbital solutions could not be meaningfully constrained for this case. Hence, in this work, we adopt the approximated lens-orbital parameters when we consider the lens-orbital effect for the best-fit models.

Table 1.  The Best-fit Models of the Ground-based Observations

		$({u}_{0}\lt 0)$		$({u}_{0}\gt 0)$
Parameter	STD	PRX	OBT+PRX	PRX	OBT+PRX
${\chi }^{2}/\mathrm{dof}$	$6501.53/(6232-7)$	$6342.13/(6232-9)$	$6227.09/(6232-11)$	$6348.94/(6232-9)$	$6240.25/(6232-11)$
t0 (HJD')	7492.261 ± 0.144	7492.636 ± 0.417	7492.478 ± 0.547	7492.188 ± 0.414	7492.595 ± 0.420
u0	0.199 ± 0.002	−0.202 ± 0.003	−0.201 ± 0.005	0.199 ± 0.003	0.207 ± 0.004
${t}_{{\rm{E}}}$ (days)	89.786 ± 0.232	88.525 ± 0.316	97.010 ± 1.345	88.550 ± 0.308	95.379 ± 1.024
s	1.120 ± 0.001	1.117 ± 0.001	1.075 ± 0.014	1.115 ± 0.001	1.092 ± 0.008
q	0.632 ± 0.009	0.664 ± 0.021	0.724 ± 0.024	0.648 ± 0.020	0.736 ± 0.024
α (rad)	5.448 ± 0.002	−5.462 ± 0.006	−5.379 ± 0.009	5.454 ± 0.006	5.367 ± 0.007
${\rho }_{\star }$ (10−2)	0.372 ± 0.007	0.375 ± 0.006	0.400 ± 0.007	0.378 ± 0.007	0.395 ± 0.007
${\pi }_{{\rm{E}},{\text{}}N}$	⋯	0.033 ± 0.004	0.382 ± 0.022	−0.038 ± 0.003	−0.475 ± 0.025
${\pi }_{{\rm{E}},{\text{}}E}$	⋯	0.013 ± 0.009	0.057 ± 0.011	0.014 ± 0.009	−0.026 ± 0.011
ds/dt (yr−1)	⋯	⋯	0.287 ± 0.120	⋯	0.090 ± 0.067
$d\alpha /{dt}$ (rad yr−1)	⋯	⋯	−1.437 ± 0.138	⋯	1.574 ± 0.121
${F}_{{\rm{S}},\mathrm{OGLE}}$	0.248 ± 0.001	0.251 ± 0.001	0.252 ± 0.001	0.250 ± 0.001	0.253 ± 0.001
${F}_{{\rm{B}},\mathrm{OGLE}}$	0.048 ± 0.001	0.045 ± 0.001	0.044 ± 0.001	0.046 ± 0.001	0.043 ± 0.001

Note. $\mathrm{HJD}^{\prime} =\mathrm{HJD}-2450000$, Abbreviations—STD: the standard, static model; PRX: the model considering the annual microlens parallax; OBT: the model considering the lens-orbital motion.

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For the ground-based models considering the microlens parallax and lens-orbital effects, the ${\chi }^{2}$ difference between the best-fit models in Table 1 is only ${\chi }^{2}({u}_{0}\gt 0)-{\chi }^{2}({u}_{0}\lt 0)=13.4$. Hence, the $({u}_{0}0\lt 0)$ solution is slightly preferred. We note that for this event there do not exist degenerate solutions caused by the close/wide degeneracy (Griest & Safizadeh 1998; Dominik 1999; An 2005) because the best-fit models have resonant caustics ($s\sim 1$).

There exist some residuals to the caustic entrance part of the light curve (HJD' ∼ 7473.9). These short-timescale systematics ($\lesssim 1$ day timescales) are not expected to affect the determination of the microlensing parameters for such a long-timescale event (${t}_{{\rm{E}}}\gg 1$ day). Photometric systematics affecting the determination of the microlens parallax or lens-orbital motion parameters typically occur in the baseline data of an event and relate to long-timescale variations, i.e., systematics with timescales $\sim {t}_{{\rm{E}}}$ (Smith et al. 2003).

These short-timescale systematics might affect the determination of these parameters if they contributed significant uncertainty to the time of the caustic entrance, which serves as a strong, fixed reference point for the determination of the lens-orbital parameters. However, despite the systematics in the KMTC data, the time of the caustic entrance is extremely well defined, especially compared to the ∼60-day timescale between the entrance and exit. In addition, it is possible to securely measure the angular source radius, i.e., ${\rho }_{* }$, from the caustic exit part of the light curve. Thus, any uncertainties in this parameter due to systematics in the caustic entrance are more than compensated for by other data. Furthermore, this measurement of ${\rho }_{* }$ means that ${t}_{* }={\rho }_{* }/{t}_{{\rm{E}}}$ is well determined, such that given the KMTA data over the peak of the caustic entrance, the timing of that caustic entrance is fixed independent of the KMTC data. Therefore, the small systematics seen in the KMTC data the night of ∼7473.9 have no influence on the model parameters. We have confirmed this by repeating the modeling excluding those data.

We conduct modeling of the combined ground and Spitzer observations. We present observed light curves with the best-fit models in Figure 1. During the modeling process, we investigated degenerate solutions caused by the "four-fold degeneracy" (Zhu et al. 2015). The degeneracies are caused by different source trajectories seen by ground and space telescopes passing over a similar lensing magnification pattern, which is reflected over the binary axis. The degenerate solutions are denoted by the combination of the signs of impact parameters as seen from the space and ground, i.e., (±, ±), according to convention (see Section 3 in Zhu et al. 2015). Under our parameterization, we can control the source trajectories by changing the sign of u₀ and the ${\pi }_{{\rm{E}},{\text{}}N}$ parameters and thus we carefully set initial values to search for these models. For this event, we find that the (−, −) and (+, +) models showed similar fits with ${\rm{\Delta }}{\chi }^{2}\sim 36.4$. However, there do not exist plausible local minima of the (+, −) and (−, +) solutions.²⁰

The observed Spitzer light curve is fragmentary and does not cover the caustic-crossing parts. Thus, we expect that "color constraint" might be important for finding the correct model including the Spitzer light curve (described in Section 5.3 of Yee et al. 2015b). To incorporate the color constraint in the modeling, we use ${(I-L)}_{18}=5.146\pm 0.124$ as derived in Section 3.3. We then introduce the "${\chi }_{\mathrm{penalty}}^{2}$" defined as

$$\begin{eqnarray}&&{\chi }_{\mathrm{penalty}}^{2}\equiv \displaystyle \frac{{\{2.5\mathrm{log}{({F}_{{\rm{S}},{\text{}}{Spitzer}}/{F}_{{\rm{S}},\mathrm{OGLE}})-(I-L)}_{18}\}}^{2}}{{\sigma }_{\mathrm{cc}}^{2}},\end{eqnarray} \tag{ 2 }$$

where ${F}_{{\rm{S}},{\text{}}{Spitzer}}$ and ${F}_{{\rm{S}},\mathrm{OGLE}}$ are the source fluxes of each observatory, (i.e., of each passband) determined from the model. The ${\sigma }_{\mathrm{cc}}$ is the uncertainty of the color constraint. The ${\chi }_{\mathrm{penalty}}^{2}$ increases as the difference between the model-fitted color and color constraint increases. Note that, we additionally increase the penalty defined as

$$\begin{eqnarray}&&{({\chi }_{\mathrm{penalty}}^{2})}^{{\prime} }={\mathrm{fac}}^{2}\times ({\chi }_{\mathrm{penalty}}^{2}),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\mathrm{if}\,\,2.5\mathrm{log}\,\left(\displaystyle \frac{{F}_{{\rm{S}},{\text{}}{Spitzer}}}{{F}_{{\rm{S}},\mathrm{OGLE}}}\right)\gt \pm \mathrm{fac}\times {\sigma }_{\mathrm{cc}}{(I-L)}_{18},\end{eqnarray} \tag{ 4 }$$

where "$\mathrm{fac}$" is a factor set as 2. It implies that we use a ${({\chi }_{\mathrm{penalty}}^{2})}^{{\prime} }$ that is four times larger than ${\chi }_{\mathrm{penalty}}^{2}$ if the color difference is larger than the $2\sigma$ level of the color constraint.

In Table 2, we present the best-fit parameters of models both with and without the color constraint. The total ${\chi }^{2}$ consists of the sum of the ${\chi }^{2}$ for the ground-based data and the ${\chi }^{2}$ for the Spitzer data; the ${\chi }^{2}$ from the color constraint is given separately. For the analysis of this event, we find that the color constraint is of minor importance to the model. In fact, when fitting without the constraint, the model parameters are recovered within $3\sigma$ of those of the best-fit model with the constraint. Particularly, the model parameters of the microlens parallax and the lens-orbital effect are recovered within the $2\sigma$ level. Because the color constraint is both an intrinsic observable and model-independent, the fact that the best-fit models satisfy this constraint serves as an independent check that they correspond to the physical system. Even though the color constraint plays a relatively minor role, we conclude the best-fit models of this event are the cases of models considering the color constraint (bold parameters in Table 2).

Table 2.  The Best-fit Models of the Combined Observations

	$(-,\,-)$		$(+,\,+)$
Parameter	w/o cc	w/cc	w/o cc	w/cc
${\chi }_{\mathrm{total}}^{2}$	6257.46	6259.74	6296.21	6296.16
${\chi }_{\mathrm{Ground}}^{2}$	6227.88	6230.48	6257.18	6256.03
${\chi }_{{\text{}}{Spitzer}}^{2}$	29.58	29.26	39.03	40.13
${\chi }_{\mathrm{penalty}}^{2}$	⋯	3.27 ($\lt 2{\sigma }_{{\bf{cc}}}$)	⋯	0.08 ($\lt 1{\sigma }_{\mathrm{cc}}$)
t0 (HJD')	7492.470 ± 0.488	7493.687 ± 0.311	7493.130 ± 0.498	7493.298 ± 0.461
u0	−0.203 ± 0.005	−0.214 ± 0.003	0.218 ± 0.005	0.220 ± 0.005
${t}_{{\rm{E}}}$ (days)	95.249 ± 1.224	93.670 ± 1.165	89.449 ± 1.036	89.395 ± 0.921
s	1.088 ± 0.012	1.104 ± 0.011	1.139 ± 0.011	1.140 ± 0.010
q	0.713 ± 0.023	0.768 ± 0.015	0.731 ± 0.027	0.741 ± 0.024
α (rad)	−5.389 ± 0.008	−5.403 ± 0.008	5.401 ± 0.007	5.401 ± 0.005
${\rho }_{\star }$ (10−2)	0.393 ± 0.007	0.396 ± 0.007	0.388 ± 0.007	0.390 ± 0.007
${\pi }_{{\rm{E}},{\text{}}N}$	0.349 ± 0.024	0.360 ± 0.027	−0.401 ± 0.034	−0.411 ± 0.023
${\pi }_{{\rm{E}},{\text{}}E}$	0.062 ± 0.011	0.047 ± 0.009	−0.002 ± 0.010	−0.004 ± 0.010
ds/dt (yr−1)	0.182 ± 0.105	0.050 ± 0.094	−0.325 ± 0.103	−0.337 ± 0.093
$d\alpha /{dt}$ (rad yr−1)	−1.238 ± 0.123	−1.133 ± 0.125	0.933 ± 0.103	0.954 ± 0.070
${F}_{{\rm{S}},\mathrm{OGLE}}$	0.252 ± 0.001	0.254 ± 0.001	0.253 ± 0.001	0.253 ± 0.001
${F}_{{\rm{B}},\mathrm{OGLE}}$	0.044 ± 0.001	0.042 ± 0.001	0.044 ± 0.001	0.043 ± 0.001
${F}_{{\rm{S}},{\text{}}{Spitzer}}$	38.967 ± 1.059	36.357 ± 0.589	28.773 ± 2.647	27.961 ± 1.692
${F}_{{\rm{B}},{\text{}}{Spitzer}}$	−7.716 ± 1.519	−4.306 ± 0.845	2.834 ± 3.339	3.726 ± 2.262

Note. $\mathrm{HJD}^{\prime} =\mathrm{HJD}-2450000$. See Section 3.3 for the definition of ${\chi }_{\mathrm{penalty}}^{2}$. The degree of freedom ($\mathrm{dof}$) of each model is (6260−11). The total numbers of ground-based and space-based data that are used for modeling are 6232 and 28, respectively. Abbreviation—cc: color constraint.

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The CMD analysis is required for two purposes in this case. First, we obtain the color constraint for the source star for this event. The color constraint is determined by model-independent regression based on the color–color diagram (described in Calchi Novati et al. 2015b). Second, we obtain the value of the angular source radius, ${\theta }_{* }$, to determine the angular Einstein ring radius, ${\theta }_{{\rm{E}}}={\theta }_{* }/{\rho }_{* }$. Thanks to good caustic coverage from KMTNet observations, it is possible to securely measure the ${\rho }_{* }$ value of this event based on the finite source effect. To achieve both purposes requires determining the position of the source on the CMD of the event. The conventional method is to use the $(V-I,I)$ CMD, but this is impossible in this case because the source suffers from severe extinction $({A}_{I}=4.9)$.

We construct an $(I-H,I)$ CMD using data from the OGLE survey and the VISTA Variables and Via Lactea Survey (VVV: Minniti et al. 2010) by cross-matching field stars within $60^{\prime\prime}$ of the source. We use the VVV H-band data (taken with a 4 m-class telescope) because the SMARTS observations (taken with a 1.3 m telescope) do not go deep enough to capture the red giant clump. Based on the CMD (see Figure 3), the centroid of the giant clump, which is the reference to measure the extinction toward the source, is ${(I-H,I)}_{\mathrm{clump}}\,=(5.26\pm 0.03,19.46\pm 0.05)$. The position of the source on the CMD is determined as follows. First, from the best-fit model (boldface in Table 2), we have ${I}_{{\rm{S}},\mathrm{OGLE}}=19.488\pm 0.004$, where ${I}_{{\rm{S}},\mathrm{OGLE}}$ $\,=\,18.0-2.5\mathrm{log}({F}_{{\rm{S}},\mathrm{OGLE}})$ $\,\pm \,1.085({\sigma }_{{F}_{{\rm{S}},\mathrm{OGLE}}}/{F}_{{\rm{S}},\mathrm{OGLE}})$. Second, based on the SMARTS data, we estimate the source color, ${({I}_{\mathrm{CTIO}}-{H}_{\mathrm{CTIO}})}_{{\rm{S}}}=2.971\pm 0.006$, using model-independent regression. We use comparison stars to convert the color to the OGLE and VVV magnitude scale. First, we estimate the difference between OGLE and CTIO, $({I}_{\mathrm{OGLE}}-{I}_{\mathrm{CTIO}})=-0.772\,\pm 0.036$. Second, we also estimate the difference between CTIO and 2MASS, $({H}_{2\mathrm{MASS}}-{H}_{\mathrm{CTIO}})=-3.036\pm 0.014$. Note that the VVV scale is identical to 2MASS. Combining this all together, the location of the source is estimated as ${(I-H,I)}_{{\rm{S}}}=(5.235\,\pm 0.039,19.488\pm 0.004)$. As shown in Figure 3, the locations of the source and the centroid of the red giant clump are almost identical.

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Based on the positions of the source and the centroid of the giant clump on the $(I-H,I)$ CMD, we can apply the conventional method (Yoo et al. 2004) to derive the de-reddened color and brightness of the source. First, we convert the color difference ${\rm{\Delta }}{(I-H)\equiv {(I-H)}_{{\rm{S}}}-(I-H)}_{\mathrm{clump}}$ to ${\rm{\Delta }}{(V-I)\equiv {(V-I)}_{{\rm{S}}}-(V-I)}_{\mathrm{clump}}$ using the color relations for giants from Bessell & Brett (1988), which gives ${\rm{\Delta }}(V-I)=0.762{\rm{\Delta }}(I-H)$. Second, using the de-reddened $(V-I)$ color and intrinsic I magnitude of the bulge giant clump as a reference (that values adopted from Bensby et al. (2013) and calculated from Nataf et al. (2013), respectively), we determine the de-reddened color and brightness of the source to be ${(V-I,I)}_{0,{\rm{S}}}=(1.04\pm 0.04,14.56\pm 0.05)$, where ${(V-I)}_{0,{\rm{S}}}={(V-I)}_{0,\mathrm{clump}}+{\rm{\Delta }}(V-I)$ and ${I}_{0,{\rm{S}}}\,={I}_{{\rm{S}}}+({I}_{0,\mathrm{clump}}-{I}_{\mathrm{clump}})$, which also gives ${A}_{I}=4.9$. We note that the distance of the source star is estimated by the relation of Nataf et al. (2013), ${D}_{{\rm{S}}}={D}_{\mathrm{GC}}/(\cos l+\sin l\times \cot \phi )\,\sim 8.5$ kpc, where ${D}_{\mathrm{GC}}=8160$ pc indicates the distance to the galactic center, l is the longitude of the source star in galactic coordinates, and $\phi =40^\circ$ is an apparent bar viewing angle, respectively. Then, the angular source radius, ${\theta }_{* }=5.56\,\pm 0.45\,\mu \mathrm{as}$, is determined by converting $(V-I)$ to $(V-K)$ based on the color–color relation in Bessell & Brett (1988), and then the angular radius of the source is determined using the color/surface-brightness relation in Kervella et al. (2004). From the value of ${\rho }_{* }$ derived from the best-fit model, we determine that the angular Einstein ring radius of this event is

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=1.41\pm 0.12\,\mathrm{mas}.\end{eqnarray} \tag{ 5 }$$

In addition, based on the location on the CMD and the intrinsic color of the source, the source spectral type is estimated to be an early K-type giant. We adopt limb-darkening coefficients appropriate for this source spectral type derived from (Claret 2000). The coefficient for the I-band is equal to ${{\rm{\Gamma }}}_{{\text{}}I}=(2u/(3-u))=0.5103$, where ${u}_{{\text{}}I}=0.6098$, assuming typical properties of an early K-type giant: effective temperature, ${T}_{\mathrm{eff}}\sim 4750$ K, metallicity, $[M/H]\sim 0.0$, turbulence velocity, ${V}_{{\rm{t}}}\lt 2.0\,\mathrm{km}\,{{\rm{s}}}^{-1}$, and surface gravity, $\mathrm{log}g\sim 2.0$.

For the color constraint, we carry out color–color regression based on the HIL color–color diagram. The relation is $(I-L)=1.238(I-H)-8.341$, where I, H, and L indicate the passbands of OGLE, VVV, and Spitzer, respectively. By adopting the color of the source, ${(I-H)}_{{\rm{S}}}=5.24$, the color constraint is ${(I-L)}_{25}=-1.854\pm 0.124$. We note that this transformation introduces an uncertainty of 0.1 mag. We also note that the arbitrary flux system of the Spitzer data has 1 flux unit equal to 25th magnitude. This system is different from the other observations, which use 1 flux unit equal to 18th magnitude, which again is an arbitrary system but one that is frequently used. Thus, we convert the color constraint from the 25th magnitude system to the 18th magnitude system by adding 7 magnitudes. Hence, we use ${(I-L)}_{18}=5.146\,\pm 0.124$ for the constraint.

Although the formal uncertainty in the blended flux as given in Table 2 is small, the blended flux is not meaningfully constrained. The uncertainty for ${F}_{{\rm{B}},\mathrm{OGLE}}$ given in Table 2 assumes that the baseline flux is known to infinite precision. However, the baseline object in the OGLE catalog has ${I}_{\mathrm{base}}=19.318\pm 0.092$. Although, ${F}_{{\rm{S}},\mathrm{OGLE}}$ is known very well from the light curve, the value of ${F}_{{\rm{B}},\mathrm{OGLE}}$ must be derived from ${F}_{{\rm{B}}}={F}_{\mathrm{base}}-{F}_{{\rm{S}}}$. Thus, the true uncertainty in ${F}_{{\rm{B}},\mathrm{OGLE}}$ is±0.03 (not 0.001), i.e., ${F}_{{\rm{B}},\mathrm{OGLE}}$ is consistent with zero.

Based on space-based observations, it is possible to conduct a test for verifying the measurement of the annual microlens parallax. In addition, as pointed out by Han et al. (2016a, 2016b), space-based observations can also provide an opportunity to resolve degenerate solutions. Thus, we conduct a test based on the Spitzer observations to verify the annual microlens parallax and lens-orbital motion effects from results of the ground-based models. In addition, we try to resolve degenerate solutions using the Spitzer observations.

In Figure 2, the red lines indicate the predicted source trajectories that should be seen by Spitzer. These predicted source trajectories are produced using ground-based models considering the annual microlens parallax and lens-orbital effects. By comparing the prediction without Spitzer and the fitting with Spitzer data, it is possible to check the measurement of the microlens parallax and lens-orbital motion. Note that the parameters of the annual and satellite microlens parallaxes are defined in the same reference frame (Calchi Novati et al. 2015a; Yee et al. 2015b; Zhu et al. 2015). As a result, it is possible to directly compare the microlens parallax values. In addition, this validation process can provide a chance to resolve the degenerate solutions.

As shown in Figure 2 (purple dots on the predicted trajectories), Spitzer observations covered only a short segment (∼26 days) compared to the total Einstein timescale of the event (∼94 days). For relatively long-timescale microlensing events, space-based observations usually cover only part of the lensing light curve due to the short observing window. As a result, this fragmentary Spitzer light curve is quite common for long-timescale lensing events. Thus, our test can provide an important example of whether it is possible to extract secure microlens parallax information from a fragmentary light curve or not.

For the case of the OGLE-2016-BLG-0168 event, we found two possible lens-parallax solutions, $(-,\,-)$ and $(+,\,+)$, out of the possible four-fold degeneracy for the microlens parallax. The ${\rm{\Delta }}{\chi }^{2}$ between those models is ∼36.4, 10.9 of which comes from ${\rm{\Delta }}{\chi }_{{\text{}}{Spitzer}}^{2}$. In addition, we found inconsistency between the prediction of the light curve covered by the Spitzer data made from the ground-based data alone for the $({u}_{0}\gt 0)$ solution and the best-fit model including Spitzer data for the $(+,+)$ solution, as indicated by the different curvatures of the Spitzer light curve seen in Figure 2. Furthermore, for the prediction of the $(+\,,+)$ case, there exist large inconsistencies in the parameters between the $({u}_{0}\gt 0)$ ground-only model and the model including Spitzer data for the $(+,\,+)$ case at the $4\sigma$ and more than $6\sigma$ levels for the microlens parallax and lens-orbital parameters, respectively (see Figure 4). Hence, considering all the clues to resolve the degeneracy, we conclude the $(-,\,-)$ model is the unique solution that describes the nature of the binary lens system of this lensing event.

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As shown in Figure 2, for the $(-,\,-)$ case, the prediction of the Spitzer light curve is almost the same as the light curve found by including the Spitzer data in the fitting. Thus, the higher-order effects measured from the ground-based light curve alone are confirmed by the Spitzer observations. Note that the prediction of the space-based light curve is dominated by the microlens parallax parameters. However, the microlens parallax parameters are strongly affected by the lens-orbital effect (Shin et al. 2012). Thus, the lens-orbital parameters are also essential factors for the successful prediction of the Spitzer light curve.

In Figure 4, we present distributions of the microlens parallax and lens-orbital parameters to clearly show the confirmation of the prediction for the ($(-,\,-)$ and $({u}_{0}\lt 0)$) case. We find that the parameters of the ground $({u}_{0}\lt 0)$ model that are used for the prediction are well matched to those of the model including Spitzer data for the $(-,\,-)$ case, i.e., within $2\sigma$ and $3\sigma$ for the microlens parallax and lens-orbital parameters, respectively.

The confirmation of the microlens parallax and the resolution of the $(-,\,-)$ to $(+,\,+)$ degeneracy show that it is possible to extract valuable information from space-based observations even though the observations are fragmentary. Although this is one specific case, it is significant because almost all space-based observations have only partial coverage of long-timescale lensing events. Thus, we frequently encounter such fragmentary light curves.

Based on the unique best-fit model and combining the information of the microlens parallax and the angular Einstein ring radius, we can determine the properties of the binary lens system according to Equations (1). In Table 3, we present the properties of the lens system. The system consists of nearly equal mass stars,

$$\begin{eqnarray}&&{M}_{1}=0.27\pm 0.03\,{M}_{\odot };\,{M}_{2}=0.21\pm 0.02\,{M}_{\odot },\end{eqnarray} \tag{ 6 }$$

with a projected separation,

$$\begin{eqnarray}&&{a}_{\perp }=2.47\pm 0.23\,\mathrm{au}.\end{eqnarray} \tag{ 7 }$$

The lens system is located 1.59 ± 0.15 kpc from us. We check the possibility of follow-up observations to confirm the properties of the lens system. Considering the relative lens-source proper motion, ${\mu }_{\mathrm{rel}}={\theta }_{{\rm{E}}}/{t}_{{\rm{E}}}=5.48\pm 0.43\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$, we expect the binary lens and source star to separate by $\sim 55\,\mathrm{mas}$ after 10 years from now. Given that JWST's resolution is $\sim 70\,\mathrm{mas}$, JWST will have difficulty making follow-up observations. However, using next-generation telescopes (30 m-class), it is quite possible that the lens will be visible for follow-up observations.

Table 3.  The Properties of the Binary Lens System

Quantity	Value
Einstein radius, ${\theta }_{{\rm{E}}}$ (mas)	1.41 ± 0.12
Total Mass, ${M}_{\mathrm{total}}$ $({M}_{\odot })$	0.48 ± 0.05
Primary Mass, M1 $({M}_{\odot })$	0.27 ± 0.03
Secondary Mass, M2 $({M}_{\odot })$	0.21 ± 0.02
Distance to lens, ${D}_{{\rm{L}}}$ (kpc)	1.59 ± 0.15
Projected separation, a⊥ (au)	2.47 ± 0.23
Geocentric proper motion, ${\mu }_{\mathrm{geo}}$ (mas yr−1)	5.48 ± 0.45
Heliocentric proper motion, ${\mu }_{\mathrm{hel}}$ (mas yr−1)	6.21 ± 0.51
Stability of system, KE/PEa	0.52

Note.

^aIf the ratio of the kinetic to potential energy of the binary lens system (KE/PE) is less then 1, then the orbital motion of binary components is physically allowed. However, values $(\mathrm{KE}/\mathrm{PE})\sim 1$ and values $(\mathrm{KE}/\mathrm{PE})\ll 1$ would require very special physical configurations and/or viewing angles. Hence, the parameters of the model considering the lens-orbital effect are quite reasonable values.

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Since we introduce orbital motion of the lens system, we check whether the best-fit orbital parameters are physically reasonable or not. Thus, we derive the ratio of kinetic to potential energy of the system to validate the stability of the lens system. The determined value $(\mathrm{KE}/\mathrm{PE})\simeq 0.5$ easily satisfies the physically bound condition $(\mathrm{KE}/\mathrm{PE})\lt 1$. Moreover, it is well away from the regimes $(\mathrm{KE}/\mathrm{PE})\sim 1$ and $(\mathrm{KE}/\mathrm{PE})\ll 1$, both of which would require special geometries and/or viewing angles. Since the orbital motion parameters are derived from quite subtle features in the light curve, they could, in principle, reflect unrecognized systematic errors in the data rather the the physical dynamics of the system. However, since the former putative explanation can yield arbitrarily high or low values of $(\mathrm{KE}/\mathrm{PE})$, whereas the latter explanation must yield $(\mathrm{KE}/\mathrm{PE})\lt 1$, and will yield $0.1\lt (\mathrm{KE}/\mathrm{PE})\lt 0.8$ for the great majority of binaries, the fact that the measured value is near the middle of this range can be regarded as evidence that binary motion, rather than systematics, is the true origin. This evidence is relatively weak for any individual case, but if the test is successfully repeated for an ensemble of events, the evidence will be much stronger. This is important in the present case because the lens is unusually close (${D}_{L}=1.6$ kpc) and the orbital motion is usually fast $(| d\alpha /{dt}| \sim 1\,\mathrm{radian}\,{\mathrm{yr}}^{-1})$. A number of the most interesting microlensing events, e.g., OGLE-2011-BLG-0417 (Shin et al. 2012), OGLE-2011-BLG-0420 and OGLE-2009-BLG-151 (Choi et al. 2013), are from such nearby lenses, which are intrinsically relatively rare but which frequently permit ground-based parallax measurements when they occur. Hence, when one of these can be verified as a physically (rather than systematics) generated light curve by several independent checks, it enhances confidence in this entire interesting class of events.

In Figure 5, we present the cumulative distribution of the "distance parameter," ${D}_{8.3}$ (see Section 5 of Calchi Novati et al. 2015a), of published microlensing events with well-measured ${\pi }_{\mathrm{rel}}(={\pi }_{{\rm{E}}}{\theta }_{{\rm{E}}})$ based on Spitzer observations. We note that the lens system of this work is the nearest one with a Spitzer distance. Assuming that two-body lenses, which dominate this sample, follow the same galactic distribution as all lenses, this distribution represents the most precise determination of the Spitzer-observed lens distance distribution. By comparing this precise distance distribution of all stellar objects to those of planets, it is possible to study differences in the planet distance distribution. Thus, this distribution is a key factor in understanding the distribution of planets in our galaxy.

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